Title

Author

Degree Type

Dissertation

Date of Award

1989

Degree Name

Doctor of Philosophy

Department

Statistics

First Advisor

Herbert David

Abstract

Suppose we have t objects C[subscript]1,...,C[subscript]t, and that objects C[subscript]i and C[subscript]j are judged pairwise in n[subscript]ij independent comparisons, for i,j = 1,...,t; i ≠ j. In the simplest of such 'paired-comparison' experiments, all pairs of objects are compared an equal number of times (i.e., all n[subscript]ij = n); much of the paired-comparison literature pertains to the design and analysis of such 'completely balanced' experiments. Yet it is often inconvenient or impractical to carry out such a design: some pairs of objects might be compared more often than others, and some pairs might not be compared at all. Most of the available methods for analysis of unbalanced paired-comparison data are parametric, in the sense that a (paired-comparison) linear model generates, for each pair of objects, the 'preference probability' [pi][subscript]ij with which C[subscript]i is preferred to C[subscript]j. The few existing nonparameteric approaches are critically examined. David (1987) proposes a simple method of scoring objects from unbalanced paired-comparison data that takes into account differences in the strength of the competition encountered by each object as well as possible differences in the number of comparisons on each pair of objects. Statistical properties of the proposed scores are developed for the general unstructured case and for special cases of partial balance, such as when objects are arranged in a group divisible design. The asymptotic distribution of these scores leads to several approximate tests of hypotheses, including a test for equality of the objects. Through some numerical examples this proposed method will be compared with the few other nonparametric method designed for unbalanced data. The approach is then extended to unbalanced ranked data. It is shown that the previous nonparametric rank approaches fail to account adequately for the aspects of unbalanced data of concern in this dissertation. Numerical examples of unbalanced ranked data illustrate the comparison between the proposed method and the existing rank methods;Reference. David, H. A. (1987). Ranking from unbalanced paired-comparison data. Biometrika 74, 2, 432-6.